Question

A manufacture has been selling 1700 television sets a week at $390 each. A market survey indicates that for each $20 rebate offered to a buyer, the number of sets sold will increase by 200 per week. a) How large rebate should the company offer to a buyer, in order to maximize its revenue? b) If the weekly cost function is 110500+130x, how should it set the size of the rebate to maximize its profit?

Answer 1

Assume the demand Q(P) is a linear function.

Q(390) = 1700

The slope = -200 /20 (The slope - demand line is always negative)

=-10

The point slope form of the equation is

Q - 1700 = -10 (P - 390)

Open the parenthesis

Q - 1700 = -10p + 3900

Q = -10p + 3900 + 1700

Q = -10p + 5600

Now solve for P

Q + 10P = 5600

10P = -Q + 5600

Divide through by 10

P = -1/10 Q + 560

Substitute Q = x

`P(X)=-(1)/(10)x\text{ + 560}`

The above is the demand function (price p as a function of units sold x).

a) The revenue function is defined as ;

R(P) = P * Q(P)

= p ( -10p + 5600)

= -10p² + 5600P

To maximaize the revenue,

Differentiate the above

R'(P) = -20P + 5600

Set R'(P)=0

-20P + 5600 =0

20P = 5600

Divide both-side by 20

P = 280

Hence, to maximize they should offer $280 to the buyers.

C)

C(x) is the cost to produce x television sets

C(Q(p)) is the cost to produce the demanded quantity